Warm-up:
We started with a quick game of online Simon Says.
It was okay, but a little clunky. Usha suggested a different kind of activity where video on means yes and video off means no. Read a series of statements. Here are a few I brainstormed: Winter is my favorite season of the year. I love ice cream. I am excited to be back in school. I want to go to college. (You probably have better examples.) End with a statement everyone can say yes to…
Small group
I decided to start with a task in small groups and generate question based on the task.
Brooke and Usha looked at the square numbers 1, 4, 9, 16, 25, and 36. They decided that 1 wouldn’t be used because there is no way to create a pair of numbers that sum to 1 if you are limited to the numbers 1 to 18. They also eliminated 36 because you can only use 18 once. Then they looked at all the possible pairs that can sum to each of these sums, looking for a combination of pairs that would use as many of the numbers 1 through 18 as possible.
They found two different solutions with 8 square-sum pairs, one with 1 and 2 left over and another with 16 and 18 left over.
Deneise, Macarena, and Sarah looked at the different ways to make 9 with the sum of two numbers. When they ran out of ways, they had used the numbers 1 through 8. They added 9 to 16 to get 25, then looked at other ways of getting 25.
They were also able to find 8 square-sum pairs, with 17 and 18 left over.
Macarena also tried another strategy in another Jamboard frame. She wrote the numbers 1 through 18 in a number line and pair numbers with arcs.
Large group
We came back together as a group to brainstorm questions related to the task:
After brainstorming these questions, we went back into breakout groups to explore.
Small Groups
Brooke and Usha looked at 1 through 13 and found that there was 1 number unpaired. They made a conjecture that 1 through 12 would have 2 left over, 1 through 11 would have l left over, 1 through 14 would have 2 left over. Their 1-12 conjecture was confirmed, but there 1-14 conjecture was wrong. They were able to pair all numbers 1-14.
Sarah, Macarena, and Deneise noticed a pattern in the sum of different sets of consecutive numbers. If you total 1 through 8, the total is 36, a square number. If you total the next 8 numbers (9-16), you get 100, another square number. Their conjecture was that they would get an even square number if they totaled the next set of numbers that paired together (17-32). However, 392 is close to an even square number (400), but it’s not square.
At the end of the meeting, we wondered if there is a way to create a visual that will show the square-sum pairs (1+8, 2+7, 3+6, etc.) but also show the total squares or rectangles that the consecutive numbers add up to. For example, the first 8 consecutive numbers add up to 36, which is a square number. 36 is also a triangular number, though, since the sum of the first n numbers is always a triangular number.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 1/2 (8 * 9) = 4 * 9 = 6 * 6 = 36
How could we show this visually?
Resources
The initial problem comes from classroom materials from Gordon Hamilton (MathPickle.com), who was inspired by Henri Picciotto. Follow the links above to download student handouts, an article about the program, and a link to a New York Times Wordplay story about the problem.
In attendance: Brooke, Deneise, Eric, Macarena, Sarah, and Usha
Note: I tried planning this meeting using only a Google Jamboard, to test how this could be useful with a class. I usually have a Google Doc with links out to a Jamboard or another whiteboard, but this requires participants to find a second link and it also means that our group notes are in a different place than our exploratory work. It’s nice to have it all in the same place.
